Astroid Pedal Curve

Author

Eric W. Weisstein
June 27, 2013

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/AstroidPedalCurve.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/AstroidPedalCurve.html.

©2013 Wolfram Research, Inc. except for portions noted otherwise

Equation

astroid=a{Cos[t]^3,Sin[t]^3};

par=FullSimplify[PedalCurve[astroid,{x0,y0},t],a>0]

Special Cases

FullSimplify[par/.Thread[{x0,y0}->{0,0}],a>0]

Cartesian

eqns={x,y}-Numerator[Together[TrigExpand[par]/.{Cos[t]->cost,Sin[t]->sint}]]

gb=Factor/@GroebnerBasis[Append[eqns,cost^2+sint^2-1],{x,y},{cost,sint},MonomialOrder->EliminationOrder]

AlgebraicDegree[gb[[1]],{x,y}]

Plot

PedalCurvePlot[{Cos[t]^3,Sin[t]^3},#,{t,0.001,2 Pi},{0.001,Pi/2,.1}]&/@{{0,0},{1,0},{2,0},{1,1},{2,2},{.2,.2},{.2,.5},{.5,0}}

Properties

par={Cos[t] (x0 Cos[t]+Sin[t] (-y0+a Sin[t])),1/2 Sin[t] (a-2 x0 Cos[t]+a Cos[2 t]+2 y0 Sin[t])};

ArcLength

(s=FullSimplify[ArcLength[par,{t,0,2π}],a>0&&(x0|y0)∈Reals])//Timing

Area

Assuming[(x0|y0)∈Reals&&a>0,Area[par,{t,0,2π}]]//FullSimplify

AreaInertiaTensor

Assuming[(x0|y0)∈Reals&&a>0,AreaInertiaTensor[par,{t,0,2π}]]//Timing

Centroid

Assuming[(x0|y0)∈Reals&&a>0,Centroid[par,{t,0,2π}]]//FullSimplify

Curvature

FullSimplify[Curvature[par,t],a>0&&(x0|y0)∈Reals]

(κ=Curvature[gb[[1]],{x,y}])//Timing

ByteCount/@κ

FullSimplify[First[κ],a>0&&(x0|y0)∈Reals]

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[par,t],a>0&&(x0|y0)∈Reals]

Slope

FullSimplify[Slope[{Cos[t] (x0 Cos[t]+Sin[t] (-y0+a Sin[t])),1/2 Sin[t] (a-2 x0 Cos[t]+a Cos[2 t]+2 y0 Sin[t])},t]]

FullSimplify[Slope[x^6-16 x^5 x0+96 x^4 x0^2-256 x^3 x0^3+256 x^2 x0^4-16 a^2 x^2 y^2+3 x^4 y^2+128 a^2 x x0 y^2-32 x^3 x0 y^2-256 a^2 x0^2 y^2+112 x^2 x0^2 y^2-128 x x0^3 y^2+3 x^2 y^4-16 x x0 y^4+16 x0^2 y^4+y^6+128 a^2 x^2 y y0-16 x^4 y y0-1024 a^2 x x0 y y0+160 x^3 x0 y y0+2048 a^2 x0^2 y y0-512 x^2 x0^2 y y0+512 x x0^3 y y0-32 x^2 y^3 y0+160 x x0 y^3 y0-128 x0^2 y^3 y0-16 y^5 y0-256 a^2 x^2 y0^2+16 x^4 y0^2+2048 a^2 x x0 y0^2-128 x^3 x0 y0^2-4096 a^2 x0^2 y0^2+256 x^2 x0^2 y0^2+112 x^2 y^2 y0^2-512 x x0 y^2 y0^2+256 x0^2 y^2 y0^2+96 y^4 y0^2-128 x^2 y y0^3+512 x x0 y y0^3-256 y^3 y0^3+256 y^2 y0^4==0,{x,y}]]

FullSimplify[Slope[x^6-16 x^5 x0+96 x^4 x0^2-256 x^3 x0^3+256 x^2 x0^4-16 a^2 x^2 y^2+3 x^4 y^2+128 a^2 x x0 y^2-32 x^3 x0 y^2-256 a^2 x0^2 y^2+112 x^2 x0^2 y^2-128 x x0^3 y^2+3 x^2 y^4-16 x x0 y^4+16 x0^2 y^4+y^6+128 a^2 x^2 y y0-16 x^4 y y0-1024 a^2 x x0 y y0+160 x^3 x0 y y0+2048 a^2 x0^2 y y0-512 x^2 x0^2 y y0+512 x x0^3 y y0-32 x^2 y^3 y0+160 x x0 y^3 y0-128 x0^2 y^3 y0-16 y^5 y0-256 a^2 x^2 y0^2+16 x^4 y0^2+2048 a^2 x x0 y0^2-128 x^3 x0 y0^2-4096 a^2 x0^2 y0^2+256 x^2 x0^2 y0^2+112 x^2 y^2 y0^2-512 x x0 y^2 y0^2+256 x0^2 y^2 y0^2+96 y^4 y0^2-128 x^2 y y0^3+512 x x0 y y0^3-256 y^3 y0^3+256 y^2 y0^4==0,{x,y}]]

TangentialAngle

FullSimplify[TangentialAngle[par,t],a>0&&(x0|y0)∈Reals]

Vector Properties

parm={Cos[t] (x0 Cos[t]+Sin[t] (-y0+a Sin[t])),1/2 Sin[t] (a-2 x0 Cos[t]+a Cos[2 t]+2 y0 Sin[t])};

assum=(a|x0|y0|t)∈Reals;

VectorLength

Assuming[assum,FullSimplify[Norm[parm]]]/.Abs[x_]^(n_/;Mod[n,2]==0):>x^n

NormalVector

Assuming[assum,FullSimplify[NormalVector[parm,t]]]/.Abs[x_]^(n_/;Mod[n,2]==0):>x^n

TangentVector

Assuming[assum,FullSimplify[TangentVector[parm,t]]]/.Abs[x_]^(n_/;Mod[n,2]==0):>x^n